Area Formulas for 2D Shapes

Why This Matters

Area formulas aren't just equations to memorize—they're the foundation for understanding how two-dimensional space works. In Honors Geometry, you're being tested on your ability to recognize relationships between shapes, understand why formulas work (not just how to use them), and apply these concepts to composite figures and real-world problems. Every area formula connects back to a few core principles: base-height relationships, decomposition into simpler shapes, and the special properties of circles.

Here's the key insight: most area formulas are actually variations of the same idea. A triangle's area comes from a rectangle. A trapezoid averages two bases. A circle uses the radius squared because area scales with the square of linear dimensions. Don't just memorize $A = \frac{1}{2}bh$—understand why that $\frac{1}{2}$ appears and when you'll see it again.

---

Quadrilaterals Built on Base × Height

The simplest area concept is multiplying two perpendicular dimensions. When a shape can be "filled" by stacking unit squares in rows and columns, base × height gives you the total count.

Rectangle

• $A = lw$—multiply length by width, the most fundamental area formula
• Perpendicular sides guarantee that length and width form a grid of unit squares
• Foundation for other formulas—triangles, parallelograms, and trapezoids all derive from this relationship

Square

• $A = s^2$—a special rectangle where length equals width
• All sides congruent means you only need one measurement to find area
• Perfect for unit analysis—demonstrates why area is measured in "square" units

Parallelogram

• $A = bh$ where $h$ is the perpendicular height, not the slant side
• Shearing principle—a parallelogram can be "cut and rearranged" into a rectangle with the same base and height
• Common exam trap—students often mistakenly use the slant side instead of the perpendicular height

---

The Half-Factor Shapes

These formulas all include $\frac{1}{2}$ because they represent portions of rectangles or products of diagonals. The half-factor appears whenever you're averaging, splitting, or working with diagonals that create symmetrical divisions.

Triangle

• $A = \frac{1}{2}bh$—exactly half of the rectangle that encloses it
• Any side can be the base—just use the perpendicular height to that specific side
• Universal formula—works for acute, right, and obtuse triangles without modification

Trapezoid

• $A = \frac{1}{2}(b_1 + b_2)h$—averages the two parallel bases, then multiplies by height
• Think of it as a "stretched" triangle—the averaging accounts for the shape widening from one base to another
• Midsegment connection—the midsegment length equals $\frac{b_1 + b_2}{2}$, so area also equals midsegment × height

Rhombus

• $A = \frac{1}{2}d_1 d_2$—uses the two diagonals, which are perpendicular bisectors of each other
• Four congruent right triangles—the diagonals divide the rhombus into triangles, each with legs $\frac{d_1}{2}$ and $\frac{d_2}{2}$
• Also a parallelogram—you can use $A = bh$ if you know a side and perpendicular height instead

---

Circles and Curved Figures

Circular area introduces $\pi$ because you're measuring curved boundaries. The key insight is that area scales with the square of the radius—double the radius, quadruple the area.

Circle

• $A = \pi r^2$—radius squared times pi, the fundamental formula for curved regions
• Why $r^2$?—area is two-dimensional, so it scales with the square of linear measurements
• Diameter trap—if given diameter, divide by 2 first; $A = \pi\left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}$

Sector of a Circle

• $A = \frac{1}{2}r^2\theta$ where $\theta$ is in radians—a "pizza slice" portion of the full circle
• Fraction of the whole—equivalent to $\frac{\theta}{2\pi} \cdot \pi r^2$, which simplifies to the formula above
• Degree version—if $\theta$ is in degrees, use $A = \frac{\theta}{360} \cdot \pi r^2$

Ellipse

• $A = \pi ab$ where $a$ is the semi-major axis and $b$ is the semi-minor axis
• Stretched circle—when $a = b = r$, the formula becomes $\pi r^2$, confirming the circle is a special ellipse
• Real-world applications—planetary orbits, oval tracks, and architectural domes all use ellipse calculations

---

Regular Polygons and the Apothem

Regular polygons require a special approach because they have multiple equal sides. The apothem—the perpendicular distance from center to side—acts like a "height" for triangular sections.

Regular Polygon

• $A = \frac{1}{2}Pa$ where $P$ is perimeter and $a$ is apothem
• Triangle decomposition—divide the polygon into $n$ congruent triangles, each with base = side length and height = apothem
• Approaching a circle—as the number of sides increases, regular polygons approximate circles, and the apothem approaches the radius

---

Quick Reference Table

---

Self-Check Questions

1. Which two quadrilateral formulas use $\frac{1}{2}$ and why does that factor appear in each case?

2. A parallelogram and a rectangle have the same base and height. How do their areas compare, and what geometric principle explains this?

3. If you triple the radius of a circle, by what factor does the area increase? What property of area does this demonstrate?

4. Compare and contrast the rhombus formula ($\frac{1}{2}d_1 d_2$) with the parallelogram formula ($bh$). When would you choose one over the other?

5. FRQ-style: A regular hexagon and a circle both have the same perimeter/circumference. Which has the greater area? Explain using the relationship between regular polygons and circles.